Algorithms And Interfaces For Real-Time Deformation Of 2D .
Algorithms and Interfacesfor Real-Time Deformationof 2D and 3D ShapesAlec JacobsonETH ZurichJune 9, 2013
Deformation is an important phasein the life of a shapecreationanalysis and manipulationconsumptionImage courtesy Romain Prévostmodeling in MayaJune 9, 2013skinning deformationAlec Jacobson3d printing# 2
Deformation is an important phasein the life of a shapecreationanalysis and manipulationconsumptionImage courtesy Romain Prévostmodeling in MayaJune 9, 2013skinning deformationAlec Jacobson3d printing# 3
User constraints drive deformation toward goalproceduralconstraintsexplicitconstraintsJune 9, 2013Alec Jacobson# 4
User constraints drive deformation toward goalproceduralconstraintsexplicitconstraintsJune 9, 2013Alec Jacobson# 5
User constraints drive deformation toward goalproceduralconstraintsexplicitconstraintsJune 9, 2013Alec Jacobson# 6
Deformation applies to images as planar shapesnon-convex “cut-out” cartoonsentire image rectangleJune 9, 2013Alec Jacobson# 7
Deformation applies to images as planar shapesnon-convex “cut-out” cartoonsentire image rectangleJune 9, 2013Alec Jacobson# 8
Real-time performance critical forinteractive design and animation2D3DJune 9, 2013Alec Jacobson# 9
Real-time performance critical forinteractive design and animation2D3DJune 9, 2013Alec Jacobson# 10
Real-time performance critical forinteractive design and animationJune 9, 2013Alec Jacobson# 11
Many previous techniques provide quality,but not speedhigh-quality solutions to nonlinearelasticity energy minimizations: secondse.g. [Botsch et al. 2006]physically accurate muscle systemsrequire off-line simulatione.g. [Teran et al. 2005]Image courtesy Joseph TeranJune 9, 2013Alec Jacobson# 12
Other techniques limit user interfaceJune 9, 2013Alec Jacobson# 13
Other techniques limit user interfacee.g. [Ju et al. 2005]June 9, 2013Alec Jacobson# 14
Other techniques limit user interfaceJune 9, 2013Alec Jacobson# 15
Other techniques limit user interfaceJune 9, 2013Alec Jacobson# 16
Other techniques limit user interfaceJune 9, 2013Alec Jacobson# 17
Each handle type has a specific task,more than just different modeling metaphorJune 9, 2013Alec Jacobson# 18
Each handle type has a specific task,more than just different modeling metaphorJune 9, 2013Alec Jacobson# 19
Each handle type has a specific task,more than just different modeling metaphorJune 9, 2013Alec Jacobson# 20
Real-time applications deserveample interfaces and high-quality deformationsSGP 2010SIGGRAPH 2011SGP 2012June 9, 2013Alec Jacobson# 21
Real-time applications deserveample interfaces and high-quality deformationsSGP 2010SIGGRAPH 2011SGP 2012SIGGRAPH Asia 2011June 9, 2013Alec Jacobson# 22
Real-time applications deserveample interfaces and high-quality deformationsSGP 2010SIGGRAPH 2011SGP 2012SIGGRAPH Asia 2011June 9, 2013Alec JacobsonSIGGRAPH 2012# 23
Real-time applications deserveample interfaces and high-quality deformationsSGP 2010SIGGRAPH 2011SGP 2012SIGGRAPH Asia 2011June 9, 2013Alec JacobsonSIGGRAPH 2012# 24SIGGRAPH 2013
Real-time applications deserveample interfaces and high-quality deformationsSGP 2010SIGGRAPH 2011SGP 2012SIGGRAPH Asia 2011June 9, 2013Alec JacobsonSIGGRAPH 2012# 25SIGGRAPH 2013
Linear Blend Skinning has known artifactsbut makes up in real-time performanceload shapeJune 9, 2013Alec Jacobson# 26
Linear Blend Skinning has known artifactsbut makes up in real-time performanceload shapeplace handlesJune 9, 2013Alec Jacobson# 27
Linear Blend Skinning has known artifactsbut makes up in real-time performanceload shapeplace handlesJune 9, 2013define weights(bind time)Alec Jacobson# 28
Linear Blend Skinning has known artifactsbut makes up in real-time performanceload shapeplace handlesJune 9, 2013define weights(bind time)Alec Jacobsondeform shape(pose time)# 29
Linear Blend Skinning has known artifactsbut makes up in real-time performanceload shapeplace handlesdefine weights(bind time)TjwjJune 9, 2013Alec Jacobson# 30
Warping and deformation tasks requiredifferent user interactionspointsskeletonsJune 9, 2013Alec Jacobsoncages# 31
Weights should be smooth,shape-aware, positive and intuitiveJune 9, 2013Alec Jacobson# 32
Weights should be smooth,shape-aware, positive and intuitiveJune 9, 2013Alec Jacobson# 33
Weights must be smooth everywhere,especially at handlesour methodextension of Harmonic Coordinates[SIGGRAPH 2011][Joshi et al. 2005]June 9, 2013Alec Jacobson# 34
Weights must be smooth everywhere,especially at handlesour methodextension of Harmonic Coordinates[SIGGRAPH 2011][Joshi et al. 2005]June 9, 2013Alec Jacobson# 35
Weights must be smooth everywhere,especially at handlesour methodextension of Harmonic Coordinates[SIGGRAPH 2011][Joshi et al. 2005]June 9, 2013Alec Jacobson# 36
Shape-awareness ensuresrespect of domain’s featuresour methodnon-shape-aware methodse.g. [Schaefer et al. 2006]June 9, 2013Alec Jacobson# 37
Shape-awareness ensuresrespect of domain’s featuresour methodnon-shape-aware methodse.g. [Schaefer et al. 2006]June 9, 2013Alec Jacobson# 38
Non-negative weights are mandatoryour methodunconstrained biharmonic[Botsch & Kobbelt 2004]June 9, 2013Alec Jacobson# 39
Non-negative weights are mandatoryour methodunconstrained biharmonic[Botsch & Kobbelt 2004]June 9, 2013Alec Jacobson# 40
Spurious extrema cause distracting artifactsunconstrainedext. of [Botsch & Kobbelt 2004]local maxlocal minJune 9, 2013Alec Jacobson# 41
Spurious extrema cause distracting artifactsbounded[SIGGRAPH 2011]local maxlocal minJune 9, 2013Alec Jacobson# 42
We explicitly prohibit spurious extremaour improved[SGP 2012]local maxlocal minJune 9, 2013Alec Jacobson# 43
Constrained optimization ensuressatisfaction of all propertiesargminwj , j 1,.,mmXj 1 shape-awarekLwj k2ext. of [Botsch & Kobbelt 2004]June 9, 2013Alec Jacobson# 44
Constrained optimization ensuressatisfaction of all propertiesargminwj , j 1,.,mZmXj 1 shape-aware smoothness mesh independence2( wj ) dV [SGP 2010]June 9, 2013Alec Jacobson# 45
Constrained optimization ensuressatisfaction of all propertiesargminwj , j 1,.,mZmXj 1 2( wj ) dV 0 wj 1, j 1, . . . , m[SIGGRAPH 2011]June 9, 2013Alec Jacobson# 46shape-awaresmoothnessmesh independencenon-negativitylocalityarbitrary handles
Constrained optimization ensuressatisfaction of all propertiesargminwj , j 1,.,mZmXj 1 2( wj ) dV rwj · ruj 0, j 1, . . . , m[SGP 2012]June 9, 2013Alec Jacobson# 47shape-awaresmoothnessmesh independencenon-negativitylocalityarbitrary handlesmonotonicity
Weights in 3D retain nice propertiesDemoJune 9, 2013Alec Jacobson# 48
Weights in 3D retain nice propertiesDemoJune 9, 2013Alec Jacobson# 49
Variational formulation allowsadditional, problem-specific constraintsJune 9, 2013Alec Jacobson# 50
Variational formulation allowsadditional, problem-specific constraintsJune 9, 2013Alec Jacobson# 51
Linear blending subspace is still too smallJune 9, 2013Alec Jacobson# 52
Linear blend skinning etc. are not richenough to stretch and twist along bonesLBSJune 9, 2013our extension[SIGGRAPH Asia 2011]Alec Jacobson# 53
Linear blend skinning etc. are not richenough to stretch and twist along bonesLBSJune 9, 2013our extension[SIGGRAPH Asia 2011]Alec Jacobson# 54
Second set of our weightsexpands skinning subspaceJune 9, 2013Alec Jacobson# 55
Stretching facilitates exaggeration,a basic principle of life-like animationJune 9, 2013Alec Jacobson# 56
Stretching facilitates exaggeration,a basic principle of life-like animationJune 9, 2013Alec Jacobson# 57
Good weights alone aren’t enoughto guarantee intuitive resultsJune 9, 2013pseudo-edges [SIGGRAPH2011]Alec Jacobsonour improved# 58method[SIGGRAPH 2012]
Good weights alone aren’t enoughto guarantee intuitive resultsJune 9, 2013pseudo-edges [SIGGRAPH2011]Alec Jacobsonour improved# 59method[SIGGRAPH 2012]
Good weights alone aren’t enoughto guarantee intuitive resultsJune 9, 2013pseudo-edges [SIGGRAPH2011]Alec Jacobsonour improved# 60method[SIGGRAPH 2012]
Many previous techniques provide quality,but not speedhigh-quality solutions to nonlinearelasticity energy minimizations: secondse.g. [Botsch et al. 2006]physically accurate muscle systemsrequire off-line simulatione.g. [Teran et al. 2005]Image courtesy Joseph TeranJune 9, 2013Alec Jacobson# 61
Other reduced models employ skinning,but still too slowneeds examples, choice of energycomplicates per-frame computation millisecondse.g. [Der et al. 2006]significant performance tuning,but grid determines complexity millisecondse.g. [McAdams et al. 2011]June 9, 2013Alec Jacobson# 62
We achieve speeds measured in microseconds80k triangles20µs per iterationJune 9, 2013Alec Jacobson# 63
We achieve speeds measured in microseconds80k triangles20µs per iterationJune 9, 2013Alec Jacobson# 64
We achieve speeds measured in microseconds80k triangles20µs per iterationJune 9, 2013Alec Jacobson# 65
We achieve speeds measured in microseconds80k triangles20µs per iterationJune 9, 2013Alec Jacobson# 66
This means speed comparable to rendering30fpsJune 9, 2013Alec Jacobson# 67
This means speed comparable to rendering30fpsJune 9, 2013Alec Jacobson# 68
User specifies subset of parameters,optimize to find remaining onesfull optimizationmesh vertex positionsJune 9, 2013Alec Jacobson# 69
User specifies subset of parameters,optimize to find remaining onesfull optimizationreduced modelskinning degrees of freedomJune 9, 2013Alec Jacobson# 70
User specifies subset of parameters,optimize to find remaining onesfull optimizationreduced modelmatrix formJune 9, 2013Alec Jacobson# 71
User specifies subset of parameters,optimize to find remaining onesfull optimizationreduced modelmatrix formreduced optimizationJune 9, 2013Alec Jacobson# 72
Enforce user constraints as linear equalitiesreduced optimizationuser constraintsfullposition onlyunconstrainedJune 9, 2013Alec Jacobson# 73
Enforce user constraints as linear equalitiesreduced optimizationuser constraintsfullposition onlyunconstrainedJune 9, 2013Alec Jacobson# 74
We reduce any as-rigid-as-possible energyfull energiesJune 9, 2013Alec Jacobson# 75
We reduce any as-rigid-as-possible energyfull energiestrianglestetrahedra“spokes”“spokes and rims”[Liu et al. 2008][Chao et al. 2010][Sorkine & Alexa 2007][Chao et al. 2010]June 9, 2013Alec Jacobson# 76
We reduce any as-rigid-as-possible energyfull energieslocal/global optimizationglobal step: fix R, minimize with respect to V0local step: fix V,0 minimize with respect to RJune 9, 2013Alec Jacobson# 77
We reduce any as-rigid-as-possible energyfull energieslocal/global optimizationprecomputeglobal step: large, sparse linear solve V0 A 1 blocal step: fix V,0 minimize with respect to RJune 9, 2013Alec Jacobson# 78
We reduce any as-rigid-as-possible energyfull energieslocal/global optimizationglobal step: large, sparse linear solve V0 A 1 blocal step: 3x3 SVD for each rotation in RJune 9, 2013Alec Jacobson# 79
We reduce any as-rigid-as-possible energyfull energieslocal/global optimizationprecomputesubstituteglobal step: small, dense linear solve T Ã 1 b̃similar to:[Huang et al. 06][Der et al. 06][Au et al. 07][Hildebrandt et al. 12]local step: 3x3 SVD for each rotation in RJune 9, 2013Alec Jacobson# 80
Direct reduction of elastic energiesbrings speed up and regularization June 9, 2013Alec Jacobson# 81
Direct reduction of elastic energiesbrings speed up and regularization full ARAP solutionJune 9, 2013Alec Jacobson# 82
Direct reduction of elastic energiesbrings speed up and regularization full ARAP solutionour smooth subspace solutionJune 9, 2013Alec Jacobson# 83
We reduce any as-rigid-as-possible energyfull energieslocal/global optimizationsubstituteglobal step: small, dense linear solve T Ã 1 b̃local step: 3x3 SVD for each rotation in Rbut #rotations full mesh discretizationJune 9, 2013Alec Jacobson# 84
We reduce any as-rigid-as-possible energyfull energieslocal/global optimizationsubstituteglobal step: small, dense linear solve T Ã 1 b̃Clusterlocal step: 3x3 SVD for each rotation in RJune 9, 2013Alec JacobsonEk# 85
Rotation evaluations may be reduced byclustering in weight spacefull energiestrianglestetrahedra“spokes”“spokes and rims”[Liu et al. 2008][Chao et al. 2010][Sorkine & Alexa 2007][Chao et al. 2010]June 9, 2013Alec Jacobson# 86
Rotation evaluations may be reduced byk-means clustering in weight spacefull energiesweight space2June 9, 2013Alec Jacobson66xj 64# 87w1 (vj )w2 (vj ).wm (vj )37775
Rotation evaluations may be reduced byclustering in weight spacefull energiesJune 9, 2013Alec Jacobson# 88
Rotation evaluations may be reduced byclustering in weight spacefull energiesr 2r 4June 9, 2013Alec Jacobsonr 64# 89
We reduce any as-rigid-as-possible energyfull energieslocal/global optimizationsubstituteglobal step: small, dense linear solve T Ã 1 b̃Clusterlocal step: 3x3 SVD for each rotation in REk#rotations #T,independent of full mesh resolutionJune 9, 2013Alec Jacobson# 90
With more and more user constraintswe fall back to standard skinningJune 9, 2013Alec Jacobson# 91
With more and more user constraintswe fall back to standard skinningJune 9, 2013Alec Jacobson# 92
With more and more user constraintswe fall back to standard skinningJune 9, 2013Alec Jacobson# 93
Extra weights expand deformation subspaceno extra weightsJune 9, 201315 extra weightsAlec Jacobson# 94
Extra weights expand deformation subspaceno extra weightsJune 9, 201315 extra weightsAlec Jacobson# 95
Subspace now rich enough forfast variational modelingfull non-linear optimizationJune 9, 2013Alec Jacobson[Botsch et al. 2006]our reduced method# 96
Subspace now rich enough forfast variational modelingfull non-linear optimizationJune 9, 2013Alec Jacobson[Botsch et al. 2006]our reduced method# 97
Subspace now rich enough forfast variational modelingfull non-linear optimizationJune 9, 2013Alec Jacobson[Botsch et al. 2006]our reduced method# 98
Subspace now rich enough forfast variational modelingfull non-linear optimizationJune 9, 2013Alec Jacobson[Botsch et al. 2006]our reduced method# 99
Subspace now rich enough forfast variational modelingfull non-linear optimizationJune 9, 2013Alec Jacobson[Botsch et al. 2006]our reduced method#100
Subspace now rich enough forfast variational modelingfull non-linear optimizationJune 9, 2013Alec Jacobson[Botsch et al. 2006]our reduced method#101
Extra weights and disjoint skeletonsmake flexible control easyFrom Cartoon Animation by Preston BlairJune 9, 2013Alec Jacobson#102
Extra weights and disjoint skeletonsmake flexible control easyJune 9, 2013Alec Jacobson#103
Extra weights and disjoint skeletonsmake flexible control easyJune 9, 2013Alec Jacobson#104
Physical dynamics also benefit from our reductionDemoJune 9, 2013Alec Jacobson#105
Physical dynamics also benefit from our reductionDemoJune 9, 2013Alec Jacobson#106
Our reduction preserves nature ofdifferent energies, at no extra costSurface ARAP0VsurfVolumetric ARAP0Vvol Msurf TJune 9, 2013Alec Jacobson Mvol T#107
Our reduction preserves nature ofdifferent energies, at no extra costSurface ARAP0VsurfVolumetric ARAP0Vvol Msurf TJune 9, 2013Alec Jacobson Mvol T#108
Volumetric deformation differsdrastically from surface-basedJune 9, 2013Alec Jacobson#109
Volumetric deformation differsdrastically from surface-basedJune 9, 2013Alec Jacobson#110
Surface artifacts prevent volume meshing,prevent volumetric processingJune 9, 2013Alec Jacobson#111
Continuous winding number for watertightsurfaces generalizes to unclean triangle meshes1w(p) 4 ZZsin( )d dSm1 Xw(p) f4 f 1[SIGGRAPH 2013]June 9, 2013Alec Jacobson#112
Generalized winding number is ideal indicatorfor graphcut segmentation of convex hull1w(p) 4 ZZsin( )d dSm1 Xw(p) f4 f 1[SIGGRAPH 2013]June 9, 2013Alec Jacobson#113
Each advance inspires new improvements,new interfacesSGP 2010SIGGRAPH 2011SGP 2012SIGGRAPH Asia 2011June 9, 2013Alec JacobsonSIGGRAPH 2012#114SIGGRAPH 2013
We derive a powerful subspacevia input shape and handle descriptions Intrinsics from shape’s geometrySemantics from handlesFully automaticNew interfacesReal-time as an invariantJune 9, 2013Alec Jacobson#115
Future outlook More semantics: large data, collisions, etc.June 9, 2013Alec Jacobson#116
Future outlookMore semantics: large data, collisions, etc. Physical interfaces June 9, 2013Alec Jacobson#117
AcknowledgementsCoauthors: Ladislav Kavan, Ilya Baran, Jovan Popović,Kaan Yücer, Alex Sorkine-Hornung, Tino Weinkauf, Denis Zorin,Olga Sorkine-HornungColleagues at NYU and ETHFriends and familyThis thesis was supported in part by NSF award IIS-0905502,ERC grant iModel (StG-2012-306877), SNF award 200021 137879,the Intel Doctoral Fellowship and a gift from Adobe Systems.June 9, 2013Alec Jacobson#118
Algorithms and Interfacesfor Real-Time Deformationof 2D and 3D ShapesPapers, videos, code: people.inf.ethz.ch/ jalec/Alec Jacobsonjacobson@inf.ethz.chJune 9, 2013
Weights should be smooth everywhere,especially at handlesOur methodExtension of HarmonicCoordinates[Joshi et al. 2005]June 9, 2013Alec Jacobson# #121
Open cages allow arbitrary line constraintsJune 9, 2013Alec Jacobson#122
Sparsity helps maintain local influenceSmoothed extension ofHarmonic Coordinates[Joshi et al. 2005]Our methodJune 9, 2013Alec Jacobson# #123
Sparsity helps maintain local influenceSmoothed extension ofHarmonic Coordinates[Joshi et al. 2005]Our methodJune 9, 2013Alec Jacobson# #124
Sparsity helps maintain local influenceSmoothed extension ofHarmonic Coordinates[Joshi et al. 2005]Our methodJune 9, 2013Alec Jacobson# #125
Sparsity helps maintain local influenceSmoothed extension ofHarmonic Coordinates[Joshi et al. 2005]Our methodJune 9, 2013Alec Jacobson# #126
Sparsity helps maintain local influenceSmoothed extension ofHarmonic Coordinates[Joshi et al. 2005]Our methodJune 9, 2013Alec Jacobson# #127
Sparsity helps maintain local influenceSmoothed extension ofHarmonic Coordinates[Joshi et al. 2005]Our methodJune 9, 2013Alec Jacobson# #128
Boundedness also helps maintain local influenceUnconstrained biharmonic[Botsch and Kobbelt 2004]June 9, 2013Alec Jacobson# #129
Boundedness also helps maintain local influenceOur methodUnconstrained biharmonic[Botsch and Kobbelt 2004]June 9, 2013Alec Jacobson# #130
Spurious local maxima also causeunintuitive responseOurmethodJune 9, 2013Extension of unconstrained biharmonic[Botsch and Kobbelt 2004]Alec Jacobson# #131
Weights propagate transformations athandles to shape in real-timeTranslateJune 9, 2013RotateAlec JacobsonScale# #132
Dropping partition of unity as explicitconstraint does not effect quality0.0150.0100.0050.000June 9, 2013original weightsAlec Jacobsonfaster weights# #133
Weights may also define an intuitive,shape-aware depth ordering in 2DzJune 9, 2013zAlec Jacobson# #134
Weights may also define an intuitive,shape-aware depth ordering in 2DJune 9, 2013Alec Jacobson# #135
Rotations at point handles may be computedautomatically based on translationsJune 9, 2013Alec Jacobson# #136
Alternative skinning methods may also takeadvantage of bounded biharmonic weightsLinear blend skinningJune 9, 2013Dual quaternion skinningAlec Jacobson# #137
Same weights can interpolate colors0x HXwj (xi )Tj xij 1June 9, 2013Alec Jacobson#138
Same weights can interpolate colorsci HXwj (xi )cjj 1June 9, 2013Alec Jacobson#139
Same functions used for color interpolationunconstrained[Finch et al. 2011]ci HXwj (xi )cjj 1Image courtesy Mark FinchJune 9, 2013Alec Jacobson#140
Same functions used for color interpolationunconstrained[Finch et al. 2011]ci HXwj (xi )cjj 1June 9, 2013Alec Jacobson#141
Same functions used for color interpolationunconstrained[Finch et al. 2011]ci HXwj (xi )cjOur[SGP 2012]j 1June 9, 2013Alec Jacobson#142
Without constraints,biharmonic unintuitively interpolates colorsUnconstrained Δ2[Finch et al. 2011]June 9, 2013Our extended method[SGP 2012]Alec Jacobson#143
Skinning weights tether optimization overmultiple-component meshesJune 9, 2013Alec Jacobson#144
Skinning weights tether optimization overmultiple-component meshesMesh-energy methodse.g. Igarashi et al. 2004,Sumner et al. 2005,Botsch et al. 2006,Sorkine and Alexa 2007,Shi et al. 2007,Solomon et al. 2011June 9, 2013Our skinning-based methodAlec Jacobson#145
Extrema distort small featuresUnconstrained[Botsch & Kobbelt 2004]weight of middle pointJune 9, 2013Alec Jacobson#146
Deformation applies to images as planar shapesnon-convex “cut-out” cartoonsentire image rectangleJune 9, 2013Alec Jacobson#147
Extra weights would expand subspace June 9, 2013Alec Jacobson#148
Extra weights would expand subspace June 9, 2013Alec Jacobson#149
Extra weights would expand subspace June 9, 2013Alec Jacobson#150
Real-time automatic degrees of freedomJune 9, 2013Alec Jacobson#151
Overlapping b-spline “bumps” in weight spaceweight space2farthest point samplingJune 9, 2013Alec Jacobson66xj 64#152w1 (vj )w2 (vj ).wm (vj )37775
Overlapping b-spline “bumps” in weight spacein weight spaceweight space2b-spline basis parameterized by distance in weight spaceJune 9, 2013Alec Jacobson66xj 64#153w1 (vj )w2 (vj ).wm (vj )37775
Overlapping b-spline “bumps” in weight spacein weight spaceweight space2b-spline basis parameterized by distance in weight spaceJune 9, 2013Alec Jacobson66xj 64#154w1 (vj )w2 (vj ).wm (vj )37775
Final algorithm is simple and FASTPrecomputation per shape rigFor a 50K triangle mesh:- Compute any additional weights12 seconds- Construct, prefactor system matrices2.7 secondsJune 9, 2013Alec Jacobson#155
Final algorithm is simple and FASTPrecomputation per shape rigFor a 50K triangle mesh:- Compute any additional weights12 seconds- Construct, prefactor system matrices2.7 secondsPrecomputation when switching constraint type6 milliseconds- Re-factor global step systemJune 9, 2013Alec Jacobson#156
Final algorithm is simple and FASTPrecomputation per shape rigFor a 50K triangle mesh:- Compute any additional weights12 seconds- Construct, prefactor system matrices2.7 secondsPrecomputation when switching constraint type6 milliseconds- Re-factor global step system22 microseconds 30 iterationsglobal: #weights by #weights linear solve[McAdams et al. 2011]local: #rotations SVDsJune 9, 2013Alec Jacobson#157
Simple drag-only interface for point handlesJune 9, 2013Alec Jacobson#158
Extrema distort small featuresBounded[Jacobson et al. 2011]weight of middle pointJune 9, 2013Alec Jacobson#159
“Monotonicity” helps preserve small featuresBounded[Jacobson et al. 2011]OurJune 9, 2013Alec Jacobson#160
Spurious extrema are unstable, may “flip”slightly larger regionJune 9, 2013Alec Jacobson#161
Spurious extrema are unstable, may “flip”slightly larger regionJune 9, 2013Alec Jacobson#162
Spurious extrema are unstable, may “flip”Unconstrained[Botsch & Kobbelt, 2004]#
Spurious extrema are unstable, may “flip”Bounded#
Lack of extrema leads to more stabilityOur#
In 2D, stretching manipulates foreshorteningLBS with rigid bonesJune 9, 2013STBS with stretchable bonesAlec Jacobson#166
Behavior depends heavily onparameterization and boundary conditions 2 x 0 x 0June 9, 2013Alec Jacobson 3 x 0#167
Behavior depends heavily onparameterization and boundary conditionsParameter Domain x 0June 9, 2013Straight Tube Boundary Conditions 2 x 0 3 x 0Alec Jacobson#168
Behavior depends heavily onparameterization and boundary conditionsTorus Boundary Conditions (but connected to straight tubes)June 9, 2013Alec Jacobson#169
Traditional winding number determinesamount of insideness10June 9, 2013Alec Jacobson#170
Traditional winding number determinesamount of insideness11001100June 9, 2013-100Alec Jacobson#1712
Winding number jumps across boundaries,harmonic otherwise21½1½0-½June 9, 2013Alec Jacobson#172
Winding number jumps across boundaries,harmonic otherwise21½1½0-½June 9, 2013Alec Jacobson#173
Sum of harmonic functions is harmonic1½0-½ June 9, 2013 Alec Jacobson #174
Winding number gracefully tendstoward indicator function1½0-½June 9, 2013Alec Jacobson#175
Hierarchical evaluation performsasymptotically betterWinding number computation time (subdivided 4June 9, 20131e5Number of input facets, mAlec Jacobson1e61e7#176
Hierarchical evaluation performsasymptotically betterWinding number computation time (SHREC 1e41e5Number of input facets, mJune 9, 2013Alec Jacobson#177
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